Sparse Group Penalties for bi-level variable selection

Many data sets exhibit a natural group structure due to contextual similarities or high correlations of variables, such as lipid markers that are interrelated based on biochemical principles. Knowledge of such groupings can be used through bi-level selection methods to identify relevant feature groups and highlight their predictive members. One of the best known approaches of this kind combines the classical Least Absolute Shrinkage and Selection Operator (LASSO) with the Group LASSO, resulting in the Sparse Group LASSO. We propose the Sparse Group Penalty (SGP) framework, which allows for a flexible combination of different SGL-style shrinkage conditions. Analogous to SGL, we investigated the combination of the Smoothly Clipped Absolute Deviation (SCAD), the Minimax Concave Penalty (MCP) and the Exponential Penalty (EP) with their group versions, resulting in the Sparse Group SCAD, the Sparse Group MCP, and the novel Sparse Group EP (SGE). Those shrinkage operators provide refined control of the effect of group formation on the selection process through a tuning parameter. In simulation studies, SGPs were compared with other bi-level selection methods (Group Bridge, composite MCP, and Group Exponential LASSO) for variable and group selection evaluated with the Matthews correlation coefficient. We demonstrated the advantages of the new SGE in identifying parsimonious models, but also identified scenarios that highlight the limitations of the approach. The performance of the techniques was further investigated in a real-world use case for the selection of regulated lipids in a randomized clinical trial.